9.7 Rocket Propulsion

Rocket propulsion relies on conservation of momentum as rockets eject propellant to gain velocity. The Tsiolkovsky equation calculates a rocket's speed in space based on its mass ratio and exhaust velocity, highlighting the importance of efficient propellant use.

Launching from Earth adds complexity due to gravity. Rockets must generate enough thrust to overcome their weight, with acceleration increasing as propellant is consumed. Careful design and trajectory planning are crucial for successful missions.

Rocket Propulsion

Conservation of momentum for rockets

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• Conservation of momentum principle states that the total momentum of a closed system remains constant over time
• Applies to systems with changing mass such as rockets where the mass decreases as propellant is ejected while velocity increases
• Rocket's momentum change is equal in magnitude and opposite in direction to the momentum change of the ejected propellant
  • Expressed mathematically as $\Delta p_{rocket} = -\Delta p_{propellant}$
• Rocket's velocity change is directly proportional to the exhaust velocity and the ratio of the change in mass to the instantaneous mass
  • Given by the equation $\Delta v_{rocket} = v_{exhaust} \frac{\Delta m}{m}$
    • $v_{exhaust}$ represents the velocity of the exhaust relative to the rocket
    • $\Delta m$ is the mass of the propellant ejected
    • $m$ is the instantaneous mass of the rocket at the time of propellant ejection

Rocket velocity in space

• Tsiolkovsky rocket equation calculates a rocket's velocity in space at a given time based on initial conditions
  • Expressed as $v_f = v_e \ln \frac{m_0}{m_f} + v_0$
    • $v_f$ is the final velocity of the rocket
    • $v_e$ is the exhaust velocity of the propellant
    • $m_0$ is the initial total mass of the rocket including propellant
    • $m_f$ is the final mass of the rocket after propellant has been expelled
    • $v_0$ is the initial velocity of the rocket
• Assumes constant exhaust velocity and a gravity-free environment (space)
• Mass ratio $R = \frac{m_0}{m_f}$ is the ratio of initial to final mass
  • Higher mass ratio leads to a greater change in velocity for a given exhaust velocity
  • Rockets designed with high mass ratios (large propellant tanks) to achieve significant velocity changes (Saturn V rocket)
• The mass ratio affects the rocket's ability to carry payload, which includes scientific instruments or satellites for space exploration

Rocket speed in Earth's gravity

• Rockets launched from Earth must overcome gravitational force
• Acceleration determined by the difference between thrust force and gravitational force divided by the instantaneous mass
  • Given by the equation $a = \frac{F_{thrust} - F_{gravity}}{m}$
    • $F_{thrust}$ is the force produced by the rocket engine
    • $F_{gravity}$ is the force due to Earth's gravity acting on the rocket
    • $m$ is the instantaneous mass of the rocket
• Velocity at a specific time calculated by integrating acceleration over time
  • Expressed as $v(t) = \int_0^t a(t) dt + v_0$
    • $v(t)$ is the velocity at time $t$
    • $a(t)$ is the acceleration as a function of time
    • $v_0$ is the initial velocity at the time of launch
• Acceleration increases as mass decreases due to propellant consumption resulting in a non-linear velocity profile
  • Rockets experience maximum acceleration near the end of the burn when mass is lowest (staged rockets)

Rocket Design and Flight Considerations

• Propulsion systems are designed to generate thrust efficiently for various mission requirements
• Aerodynamics plays a crucial role in rocket design, affecting stability and performance during atmospheric flight
• The trajectory of a rocket is carefully planned to optimize fuel consumption and achieve desired orbital parameters